TITLE: On the definition of an exact sequence in an abelian category
QUESTION [13 upvotes]: I am slightly confused about the notion of exactness in a general abelian category (I want to stay clear of anything related to the Mitchell embedding theorem). Here are two definitions that I have seen:
A sequence $A \xrightarrow{f} B \xrightarrow{g} C$ is exact at $B$ if

$\operatorname{Im} f = \operatorname{Ker} g$, in the sense that they are isomorphic as subobjects;
or
$gf = 0$ and the canonical morphism $\operatorname{Im} f \rightarrow \operatorname{Ker} g$ is an isomorphism.

Could someone please shed some light on the equivalence between these two notions? A priori, one seems more general/stronger than the other.
Thanks for your time.

REPLY [8 votes]: $\operatorname{im}f$ is the kernel of the cokernel, hence is naturally equipped with a morphism $\to B$ and this is a monomorphism (as is always the case for kernels).
If we assume (1), then by the isomorphism of subobjects, $\operatorname{im}f$ factors over $\ker g$ (and vice versa), hence $\operatorname{im}f\to C$ is the zero morphism and finally $g\circ f=0$. To put it differently, the given isomorphism $\operatorname{im}f\to\ker g$ is the canonical morphism obtained from the fact the $\operatorname{im}f\to C$ is zero and so this is an isomorphism.
If we assume (2), then the canonical $\operatorname{im}f\to\ker g$ obtained from $g\circ f=0$ by definition is a factorization of $\operatorname{im} g\to B$  over $\ker g\to B$ and similar for its inverse, so we have $(1)$